Closure of Subset of Closed Set of Topological Space is Subset/Proof 1

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Theorem

Let $T$ = $\struct {S, \tau}$ be a topological space.

Let $F$ be a closed set of $T$.

Let $H \subseteq F$ be a subset of $F$.

Let $H^-$ denote the closure of $H$.


Then $H^- \subseteq F$.


Proof

\(\displaystyle H\) \(\subseteq\) \(\displaystyle F\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle H^-\) \(\subseteq\) \(\displaystyle F^-\) Topological Closure of Subset is Subset of Topological Closure
\(\displaystyle \) \(=\) \(\displaystyle F\) Set is Closed iff Equals Topological Closure

$\blacksquare$